A common refrain is that Roe v. Wade, the Supreme Court decision that denied states the ability to ban early-term abortion, benefits Republicans because it allows Republican leaders to oppose abortion while knowing they won't have to actually vote to restrict it.
Ramesh Ponnuru argued recently that the conventional wisdom is wrong. In fact, repealing Roe would harm the Democrats as much as the Republicans because it would require them to choose between placating their hard-left base, which supports little to no restrictions on abortion, or majority of the country, which is in favor of numerous limitations.
Ponnuru also makes a good point about how overturning Roe would probably effectively take the abortion question out of hands of the federal government. A national solution would be a bad idea and would not get support from enough senators to pass.
Let's say Roe is struck down tomorrow. Congress could then impose a compromise solution. Who would support a compromise though? Senators in solid blue or red states have constituents that usually have strong opinions on abortion one way or the other. If, for example, Charles Schumer decided to support a bill that protected abortions in the first trimester but only allowed them in subsequent trimesters in cases or rape, incest, or to protect the mother's health, his political standing in NY would be harmed because most New Yorkers want more liberal laws. Same problem for John Thune of South Dakota, home of the most draconian abortion laws.
So Senators in red or blue states are unlikely to support a national compromise. That leaves only the members of the swing states who might support a national compromise solution. But are there enough senators from swing states to make that happen? Unlikely. Moreover, even if there were, abortion is a single (although politically charged) issue. Rick Santorum is ardently pro-life, but he's been a senator for 12 years (a reign that might end this September) in Pennsylvania, a swing state. Although the center can now vote for him because of Roe and know he won't be able to ban abortion (an option that won't exist if Roe is struck down), it's unlikely that the abortion question will cause a great shift in swing states of voting against pro-life candidates.
The one group that might accept a compromise are those Senators with designs on the White House. They need to move to the center in order to win. But first of all, how many Senators are running for President? I do not believe they, plus swing state Senators, could garner enough votes to pass abortion legislation. But an even bigger problem is that Senators know they have to win their primary first, where extremist voters hold all the power. A national solution that is unfavorable will make winning the primary difficult, and should serve as a disincentive for Senators voting for a compromise.
The reality is that the federal government will be unable to solve the abortion question and it will be properly decided by the states. If abortion is removed from the table, a repeal of Roe will have little effect on federal politics and will not hurt or help either party nationally.
But will it affect state legislatures by causing pro-choice voters to vote Democrat? It might cause a few states to turn blue, but the change will be minimal. Again, most states are either red or blue and will be unaffected (in fact those states will probably become more solidly red or blue because people might feel like the state can actually get something done in abortion realm). It might change in the swing states, but most legislators in the swing states are probably not strongly pro-life or pro-choice anyway, because they represent a subsection of the population. At worst we'll see less extremists in swing states and more moderate Republicans or Democrats. Not a major shift, and certainly not the end of the world.
Personally I'd take a return to a sensible jurisprudence in exchange for losing a few seats in state legislatures any day.